Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4122169 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15257 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,9,7)(8,11,12,10)(14,17)(15,19)(21,22,26,28)(23,25,27,24);(6,7)(10,11)(13,14)(15,20)(16,17)(18,19)(21,23,26,27)(22,24,28,25);(8,12)(10,11)(14,17)(18,20)(21,24,26,25)(22,27,28,23);(1,2)(3,4)(5,7,9,6)(8,10,12,11)(13,15,16,19)(14,18,17,20)(21,25,26,24)(22,23,28,27);(5,8)(6,10)(7,11)(9,12)(13,14)(15,20)(16,17)(18,19)(21,23)(22,25)(24,28)(26,27)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=g,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef92 conjugacy classes — view the full character table.
| maximal subgroup of | #72232335, #72232370, #72232376, … |
| maximal normal subgroup of | #72232335, #72232370, #72232376, … |
| maximal quotient of | #72323504, #72329870, #72330944 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4029985);